Abstract

By using Mawhin’s continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of at least periodic solutions for a kind of -species Lotka-Volterra food-chain system with delays and impulses on time scales. One example is given to illustrate the effectiveness of our results.

1. Introduction

The food-chain phenomenon is universal and interesting in ecosystem. It is one of important methods to investigate this phenomenon by establishing the model of differential equations. The importance of studying the dynamics of food chain was pointed by Rosenzweig. In his famous paper on the paradox of enrichment [1], he wrote “Man must be very careful in attempting to enrich an ecosystem in order to increase its food yield. There is a real chance that such activity may result in decimation of the food species that are wanted in greater abundance.” Hereafter, many scholars investigated various kinds of food-chain systems (see [2–9]).

In [10], Li and Zhao have studied the existence of multiple positive periodic solutions for the following Lotka-Volterra food-chain system:
As we know, in population dynamics, many evolutionary processes experience short-time rapid chance after undergoing relatively long sooth variation. Examples include stocking of species and annual immigration. Incorporating these phenomena give us impulsive differential equations. For the theory of impulsive differential equations, we refer the reader to [11, 12]. In addition, there are few results on the existence of multiple periodic solutions for the delay food-chain system with impulsive effects in literatures. This motivates us to modify system (1.1) to the form
where . is a strictly increasing sequence with and . is the th species population density. is the growth rate of the only producer. and stand for the th species intraspecific competition rate and harvesting rate, respectively. is the th species death rate. represents the th species predation rate on the th species. stands for the transformation rate from the th species to the th species. stands for the time-lag in the process of the th species predation rate on the th species. stands for the time-lag in the process of transformation from the th species to the th species. represents the time-lag in the process of intraspecific competition. In addition, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (e.g., seasonal effects of weather, food supplies, mating habits, etc.), which leads us to assume that , , , , and are all continuous -periodic functions. For impulsive effects, we further assume that there exists a such that and .

However, dynamics in each equally spaced time interval may vary continuously. So, it may be more realistic to assume that the population dynamics involves the hybrid discrete-continuous processes. For example, Gamarra and Solé pointed out that such hybrid processes appear in the population dynamics of certain species that feature nonoverlapping generations: the change in population from one generation to the next is discrete and so is modeled by a difference equation, while within-generation dynamics vary continuously (due to mortality rates, resource consumption, predation, interaction, etc.) and thus are described by a differential equation [13]. The theory of calculus on time scales (see [14, 15] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [16] in order to unify continuous and discrete analysis, and it has become an effective approach to the study of mathematical models involving the hybrid discrete-continuous processes. This motivates us to consider Lotka-Volterra food-chain system with delays and impulses on time scales of the following form:
where is a -periodic time scale. is a constant. is a strictly increasing sequence with and . , , , , , , , , are rd-continuous -periodic functions. For impulsive effects, we further assume that there exists a integer such that and . We denote by , . Without loss of generality, we also assume that .

Remark 1.1. In (1.3), let , . If (the set of all real numbers), then (1.3) reduces to (1.2).

To the best our knowledge, few papers have been published on the existence of multiple periodic solutions for this model. Our main purpose of this paper is by using Mawhin’s continuation theorem of coincidence degree theory [17], to establish the existence of at least periodic solutions for system (1.3). For the work concerning the multiple existence of periodic solutions of periodic population models which was done using coincidence degree theory, we refer to [18–25].

The organization of the rest of this paper is as follows. In the Section 2, we will introduce some basic notations and lemmas which are used in what follows. In Section 3, by employing the continuation theorem of coincidence degree theory and the skills of inequalities, we establish the existence of at least periodic solutions of system (1.3). In the final Section, one example is given to illustrate the effectiveness of our results.

2. Preliminaries on Time Scales

In this section, we briefly recall some basic definitions and lemmas on time scales which are used in what follows. For more details, one can see [14–16].

Let be a nonempty closed subset (time scale) of . The forward and backward jump operators , and the graininess are defined, respectively, by

A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .

Let . Throughout this paper, the time scale is assumed to be -periodic, that is, implies . In particular, the time scale under consideration is unbounded above and below.

Definition 2.1. A function is called regulated provided its right-side limits exist (finite) at all right-side points in and its left-side limits exist (finite) at all left-side points in .

Definition 2.2. A function is called rd-continuous provided it is continuous at right-dense point in and its left-side limits exist (finite) at left-dense points in . The set of rd-continuous functions will be denoted by .

Definition 2.3. Assume and . Then we define to be the number (if it exists) with the property that given any there exists a neighborhood of (i.e., for some ) such that
for all . we call the delta (or Hilger) derivative of at . The set of functions that are differentiable and whose derivative is rd-continuous is denoted by .

If is continuous, then is rd-continuous. If is rd-continuous, the is regulated. If is delta differentiable at , then is continuous at .

Lemma 2.4. Let be regulated, then there exists a function which is delta differentiable with region of differentiation such that

Definition 2.5. Assume is a regulated function. Any function as in Lemma 2.4 is called a -antiderivative of . We define the indefinite integral of a regulated function by
where is an arbitrary constant and is a -antiderivative of . We define the Cauchy integral by
A function is called an antiderivative of provided

Lemma 2.6. If , and , then (i); (ii) if for all , then ; (iii) if on , then .

For convenience, we now introduce some notation to be used throughout this paper. Let
where is an -periodic real function, that is, for all .

Lemma 2.7 (see [26]). Assume that is a function on such that (i) is uniformly bounded on , (ii) is uniformly bounded on . Then there is a subsequence of which converges uniformly on .

3. Existence of at Least Periodic Solutions

In this section, by using Mawhin’s continuation theorem and the skills of inequalities, we will show the existence of periodic solutions of (1.3). To do so, we need to make some preparations.

Let and be real normed vector spaces. Let be a linear mapping and be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if dim and is closed in . If is a Fredholm mapping of index zero, then there exists continuous projectors and such that and , and , . It follows that is invertible and its inverse is denoted by . If is a bounded open subset of , the mapping is called -compact on , if is bounded and is compact. Because is isomorphic to , there exists an isomorphism .

Lemma 3.1 (see [17]). Let be a Fredholm mapping of index zero and let be -compact on . Assume (a) for each , every solution of is such that ; (b) for each ; (c). Then has at least one solution in .

For simplicity, we need to introduce some notations as follows:
Throughout this paper, we need the following assumptions:, , . The following results will play an important role in the proof of our main result.

Lemma 3.2. Let be a -periodic time scale. is a strictly increasing sequence with and . , , . Suppose be a -periodic function which is -continuous for , exists the right limit and left limit in the sense of time scales. Then

Proof. First, take such that . Next, take such that . Then we have
Noticing that , similar to the above, we can get
From (3.3) and (3.4), we get
that is
which implies that (3.2) holds. This completes the proof.

Lemma 3.3. Let , , and , for the functions and , the following assertions hold. (1) and are monotonically increasing and monotonically decreasing on the variable , respectively. (2) and are monotonically decreasing and monotonically increasing on the variable , respectively. (3) and are monotonically decreasing and monotonically increasing on the variable , respectively.

Proof. In fact, for all , , , we have
By the relationship of the derivative and the monotonicity, the above assertions obviously hold. The proof of Lemma 3.3 is complete.

Lemma 3.4. Assume that , , and hold, then we have the following inequalities: (1);
(2);
(3).

Proof. (1) Since
Applying Lemma 3.3, we have
Thus, we have , . The proof of (1) of Lemma 3.4 is complete. For (2) of Lemma 3.4, we similarly have
which imply that , ; that is, the inequality (2) of Lemma 3.4 holds. Let us now to prove the inequality (3) of Lemma 3.4. In fact, since
we have
If , then , and
If , then , and
So, we drive . From the above, we have completed the proof of Lemma 3.4.

Theorem 3.5. Assume that , , and hold. Then system (1.3) has at least -periodic solutions.

Proof. Let is a piecewise continuous map with first-class discontinuity points in , and at each discontinuity point, it is continuous on the left}. Take
and define
Then both and are Banach spaces. Let
where
Obviously,
Since is closed in , , , and , we know that is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given by
Then
where
By the Lebesgue convergence theorem, and are continuous. Next, we show that maps bounded sets into relatively compact sets. Let be an arbitrary bounded set in , then there exists a number such that for any . We prove that is relatively compact. In fact, for any and , we have
Since , it follows from the periodicity that there exists a constant such that . For any and , , we have
Similarly, there has a constant such that
It follows from the definition of that the right limit of exists at each for . . Without loss of generality, here, we can assume that and , for if or , we only need to add intervals or into the consideration. Consider the following functions:
where . Then on each interval , , and are uniformly bounded on , . By Lemma 2.7, there exists a subsequence of converges uniformly on . Similarly, there is a subsequence of converges uniformly on . Repeating such a process, we can obtain that there is a subsequence of converges uniformly on . Obviously,